Year 12 Complex Numbers: Division and conjugates

Division of complex numbers looks awkward at first because standard form requires the denominator to be real, not complex. The tool that fixes this is the conjugate. By multiplying both numerator and denominator by the conjugate of the denominator, the denominator becomes real and the fraction can then be simplified into the form $a + b i$ . This topic is really an application of multiplication, but it relies on using that multiplication strategically.

To divide by

a + b i

, multiply top and bottom by

a - b i

so the denominator becomes real.

Subtopic 1: What a conjugate does

The conjugate of $a + b i$ is $a - b i$ . When these are multiplied together, the imaginary cross-terms cancel and the result is $a^{2} + b^{2}$ , which is real. That is why conjugates are useful in division: they remove the imaginary part from the denominator without changing the value of the fraction, provided both numerator and denominator are multiplied by the same conjugate.

Worked example 1

Problem: Simplify

1 / (2 + i)

The conjugate of $2 + i$ is $2 - i$ .
Multiply top and bottom by $2 - i$ : $(2 - i) / ((2 + i) (2 - i))$ .
Simplify the denominator: $(2 + i) (2 - i) = 4 - i^{2} = 5$ .
So the expression becomes $(2 - i) / 5$ .
Write this in standard form: $2 / 5 - (1 / 5) i$ .

Answer:

1 / (2 + i) = 2 / 5 - (1 / 5) i

Subtopic 2: Dividing one complex number by another

When the numerator is also complex, the same process still works. Multiply both numerator and denominator by the conjugate of the denominator, expand carefully, simplify, and then separate the real and imaginary parts. The method is consistent even when the arithmetic becomes longer.

Worked example 2

Problem: Simplify

(3 + 2 i) / (1 - i)

The conjugate of $1 - i$ is $1 + i$ .
Multiply top and bottom by $1 + i$ : $((3 + 2 i) (1 + i)) / ((1 - i) (1 + i))$ .
Expand the numerator: $3 + 3 i + 2 i + 2 i^{2} = 3 + 5 i - 2 = 1 + 5 i$ .
Expand the denominator: $1 - i^{2} = 2$ .
So the result is $(1 + 5 i) / 2 = 1 / 2 + (5 / 2) i$ .

Answer:

(3 + 2 i) / (1 - i) = 1 / 2 + (5 / 2) i

Why this is called rationalising the denominator

In school algebra, rationalising often means removing a square root from a denominator. With complex numbers, the same spirit applies: you are removing the non-real structure from the denominator so the expression can be written more cleanly. The denominator after conjugate multiplication becomes a real number, which means the final result can be separated into the standard complex form much more easily.

Worked example 3

Problem: Explain why

(4 + 3 i) (4 - 3 i)

is real.

The two factors are conjugates.
Expand: $16 - 12 i + 12 i - 9 i^{2}$ .
The imaginary terms cancel.
Since $i^{2} = - 1$ , the result is $16 + 9 = 25$ .

Answer: the product is real because conjugates cancel the imaginary cross-terms.

What division reveals about complex structure

Division is a good test of whether the multiplication rules are understood properly. It also highlights the structural role of conjugates. These are not just symbolic opposites; they are precisely the partners that turn a complex denominator into a real number. Once that idea is secure, division stops feeling like a special case and becomes a predictable extension of the multiplication topic.

Common traps

Multiplying by the denominator again instead of by its conjugate.
Using the wrong conjugate sign.
Expanding the numerator correctly but simplifying the denominator incorrectly.
Leaving the answer as a fraction over a complex denominator instead of standard form.

Revision focus

In revision, say the conjugate step explicitly before doing any arithmetic. That keeps the method stable: identify the denominator, write its conjugate, multiply top and bottom, then simplify. Students who rush straight into expansion often create avoidable errors at the start.

It is also useful to check the denominator independently. If the denominator after conjugate multiplication is not purely real, the conjugate step or the expansion has gone wrong. That single check catches a large number of mistakes very quickly.

Practice links

Basic operations on complex numbers